semirings: two monoids as one, in holy haskimony

[ algebra, bsd3, data, data-structures, library, math, mathematics, maths ] [ Propose Tags ]

In mathematics, a semiring is an algebraic structure consisting of a set together with two binary operations, one commutative and one associative. A semiring has two identity elements respective to its operations. Thus a semiring can be seen as a combination of two monoids, a commutative monoid and an associative monoid.


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Versions 0.0.0, 0.1.0, 0.1.1, 0.1.2, 0.1.3.0, 0.2.0.0, 0.2.0.1
Change log CHANGELOG.md
Dependencies base (>=4.5 && <5), containers (>=0.5.4 && <0.6.1.0), hashable (>=1.1 && <1.3), integer-gmp, nats (>=0.1 && <2), semigroups, tagged, transformers, unordered-containers (==0.2.*), vector (>=0.7 && <0.13.0.0) [details]
License BSD-3-Clause
Copyright Copyright (C) 2018 chessai
Author chessai
Maintainer chessai <chessai1996@gmail.com>
Category Algebra, Data, Data Structures, Math, Maths, Mathematics
Home page http://github.com/chessai/semirings
Bug tracker http://github.com/chessai/semirings/issues
Source repo head: git clone git://github.com/chessai/semirings.git
Uploaded by chessai at Mon Jul 30 17:02:09 UTC 2018
Distributions NixOS:0.2.0.1
Downloads 309 total (131 in the last 30 days)
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Status Docs available [build log]
Last success reported on 2018-07-30 [all 1 reports]
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Readme for semirings-0.2.0.1

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semirings

Haskellers are usually familiar with monoids and semigroups. A monoid has an appending operation <> or mappend and an identity element mempty. A semigroup has an append <>, but does not require an mempty element.

A Semiring has two appending operations, 'plus' and 'times', and two respective identity elements, 'zero' and 'one'.

More formally, A semiring R is a set equipped with two binary relations + and *, such that:

  • (R, +) is a commutative monoid with identity element 0:
    • (a + b) + c = a + (b + c)
    • 0 + a = a + 0 = a
    • a + b = b + a
  • (R, *) is a monoid with identity element 1:
    • (a * b) * c = a * (b * c)
    • 1 * a = a * 1 = a
  • Multiplication left and right distributes over addition
    • a * (b + c) = (a * b) + (a * c)
    • (a + b) * c = (a * c) + (b * c)
  • Multiplication by '0' annihilates R:
    • 0 * a = a * 0 = 0

*-semirings

A *-semiring (pron. "star-semiring") is any semiring with an additional operation 'star' (read as "asteration"), such that:

  • star a = 1 + a * star a = 1 + star a * a

A derived operation called "aplus" can be defined in terms of star by:

  • star :: a -> a
  • star a = 1 + aplus a
  • aplus :: a -> a
  • aplus a = a * star a

As such, a minimal instance of the typeclass 'Star' requires only 'star' or 'aplus' to be defined.

use cases

semirings themselves are useful as a way to express that a type is both a commutative and associative monoid.

*-semirings are useful in a number of applications; such as matrix algebra, regular expressions, kleene algebras, graph theory, tropical algebra, dataflow analysis, power series, linear recurrence relations.

Some relevant (informal) reading material:

http://stedolan.net/research/semirings.pdf

http://r6.ca/blog/20110808T035622Z.html

https://byorgey.wordpress.com/2016/04/05/the-network-reliability-problem-and-star-semirings/

additional credit

Some of the code in this library was lifted directly from the Haskell library 'semiring-num'.